1. What does the Fundamental Theorem of Calculus say?

2. Let . What is f´(x) ?

3. a) Suppose . What are f´(x) and f´´(x) ?

b) Let . Compute g´(x).

4. We are given a differentiable, odd function f defined on [-3,3] that has zeros at x = -2,0, and 2 (and nowhere else) and critical points at x = -1 & 1 (and nowhere else). Also we know that
f(-1) = 1. Define a new function F on [-3,3] by the formula

a) Plot f.

b) Find the value of F(-2), F(2), and an upper and lower bound on F(0).

c) Find the critical points and inflection points of F on [-3,3].

d) Plot f and F on the same axes from [-3,3].

e) Interpret the points found in part c in terms of the graphs of both f and F.

5. a) If for all numbers a and b, what is ?

b) If , what is ?

c) If for all numbers a and b, what might f(x) and u(x) be? Are they unique?

6. Explain geometrically why:

a) if f is even

b) if f is odd

7.

a) List the steps, as succinctly as possible, in the proof of the Fundamental Theorem of Calculus.

b) Give graphical interpretations of both parts of the Fundamental Theorem.

c) In light of the second part of the Fundamental Theorem, are Riemann sums no longer a

useful means for evaluating definite integrals? Why or why not? Hint: Think about how

to evaluate .